Asymptotics for Christoel functions

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1 Asymptotics for Christoel functions Tivadar Danka University of Szeged, Bolyai Institute September 23th, 2015, BME analysis seminar T. Danka (University of Szeged) Christoel functions / 37

2 Christoel functions Let µ be a nite Borel measure on C such that K := supp(µ) is compact and K contains innitely many points. Denition (Christoel functions) The n-th Christoel function associated with µ is dened as Pn (z) 2 λ n (µ, z 0 ) := inf deg(p n) n P n (z 0 ) dµ(z), 2 where the inmum is taken for all polynomials of degree at most n. In other words: λ n (µ, z 0 ) is the norm of the evaluation functional for z 0 in L 2 (µ) P n. L 2 (µ) P n P n P n (z 0 ) T. Danka (University of Szeged) Christoel functions / 37

3 Christoel functions p n (z, µ) = p n (z): n-th orthonormal polynomial with respect to µ K n (z, w, µ) = K n (z, w): Christoel-Darboux kernel dened as K n (z, w) := n p k (z)p k (w). k=0 K n is sometimes called reproducing kernel for polynomials of degree n, since if Π n is an arbitrary polynomial of degree n, then Π n (z) = K n (z, w)π n (w)dµ(w) = K n (z, ), Π n ( ). It is known that λ n (z, µ) = Kn (z, w) 2 K n (z, z) 2 dµ(w) = 1 K n (z, z). T. Danka (University of Szeged) Christoel functions / 37

4 Szeg 's theorem Szeg obtained the following theorem in the early 20th century. Theorem (Szeg ) Let µ be a nite Borel measure supported on the unit circle T such that Then µ is a.c. with dµ(e it ) = w(e it )dt and π π log w(eit )dt > holds. (This is called the Szeg condition.) 1 2π lim λ n(µ, z) = (1 z 2 ) exp n ( 1 2π π Re π ) [ e it + z ] e it log w(e it )dt, z < 1. z T. Danka (University of Szeged) Christoel functions / 37

5 Szeg 's theorem What about points on the unit circle? Theorem Let µ be a nite Borel measure supported on the unit circle with dµ(e it ) = w(e it )dt + dµ s (e it ), t [ π, π). Then lim λ n(µ, z 0 ) = µ({z 0 }), z 0 = 1. n In this case, we want to obtain information about the rate of convergence. T. Danka (University of Szeged) Christoel functions / 37

6 Theorem of A. Máté, P. Nevai and V. Totik Theorem (Máté, Nevai, Totik, 1991) Let µ be a nite Borel measure supported on the unit circle with dµ(e it ) = w(e it )dt + dµ s (e it ), t [ π, π). If the Szeg condition holds, then 1 π log w(e it )dt > 2π π lim nλ n(µ, e it ) = 2πw(e it ), n t [ π, π) almost everywhere. T. Danka (University of Szeged) Christoel functions / 37

7 Theorem of A. Máté, P. Nevai and V. Totik Theorem (Máté, Nevai, Totik, 1991) Let µ be a nite Borel measure supported on the unit circle with dµ(e it ) = w(e it )dt + dµ s (e it ), t [ π, π). If the Szeg condition holds, then 1 π log w(e it )dt > 2π π lim nλ n(µ, e it ) = 2πw(e it ), n t [ π, π) almost everywhere. Theorems of these type can be generalized in two ways. Study measures with more general support. Study measures supported on the unit circle, but with weaker conditions. T. Danka (University of Szeged) Christoel functions / 37

8 Theorem of A. Máté, P. Nevai and V. Totik Theorem (Máté, Nevai, Totik, 1991) Let µ be a nite Borel measure supported on the unit circle with dµ(e it ) = w(e it )dt + dµ s (e it ), t [ π, π). If the Szeg condition holds, then 1 π log w(e it )dt > 2π π lim nλ n(µ, e it ) = 2πw(e it ), n t [ π, π) almost everywhere. Theorems of these type can be generalized in two ways. Study measures with more general support. Study measures supported on the unit circle, but with weaker conditions. (Or study measures with more general support and weaker conditions.) T. Danka (University of Szeged) Christoel functions / 37

9 An outline of results The known results can be summarized like this. Nice measures on [ 1, 1] or T Locally nice measures on [ 1, 1] or T Locally nice measures on compact sets of R Locally nice measures on curves lying in C Measures with power type singularity on compact sets of R Measures with power type singularity on curves lying in C T. Danka (University of Szeged) Christoel functions / 37

10 Local version of Máté-Nevai-Totik Let I [ π, π) be a proper subinterval. What happens if, instead of π π log w(eit )dt >, we only assume I log w(e it )dt >? (This is called local Szeg condition.) In this case, a global condition is needed on the measure. Denition (Regularity in the Stahl-Totik sense) µ is said to be regular in the sense of Stahl and Totik, or µ Reg, if for every sequence of polynomials {P n } n=0 we have lim sup n ( P n (z) P n L2 (µ) ) 1/ deg(pn) 1, z supp(µ). T. Danka (University of Szeged) Christoel functions / 37

11 Local version of Máté-Nevai-Totik The local version of the previous theorem of Máté, Nevai and Totik says the following. Theorem (Máté, Nevai, Totik, 1991) Let µ be a nite Borel measure regular in the sense of Stahl-Totik supported on the unit circle with dµ(e it ) = w(e it )dt + dµ s (e it ), t [ π, π). If for some interval I [ π, π) the local Szeg condition 1 log w(e it )dt > 2π I holds, then lim nλ n(µ, e it ) = 2πw(e it ), n t I almost everywhere. T. Danka (University of Szeged) Christoel functions / 37

12 Real version of Máté-Nevai-Totik A. Máté, P. Nevai and V. Totik have obtained analogous results for measures supported on the interval [ 1, 1]. Theorem (Máté, Nevai, Totik, 1991) Let µ be a nite Borel measure regular in the sense of Stahl-Totik supported on [ 1, 1] with dµ(x) = w(x)dx + dµ s (x). If for some interval I [ 1, 1] the local Szeg condition log w(x) π dx > 2 1 x holds, then I lim nλ n(µ, x) = π 1 x 2 w(x), n x I almost everywhere. T. Danka (University of Szeged) Christoel functions / 37

13 Concepts from potential theory Let µ be a nite Borel measure on R and suppose that K := supp(µ) is compact. How can we describe lim n nλ n(µ, x) when K can be any compact set of R, not just [ 1, 1]? T. Danka (University of Szeged) Christoel functions / 37

14 Concepts from potential theory Let µ be a nite Borel measure on R and suppose that K := supp(µ) is compact. How can we describe lim n nλ n(µ, x) when K can be any compact set of R, not just [ 1, 1]? To do this, we need concepts from potential theory. T. Danka (University of Szeged) Christoel functions / 37

15 Concepts from potential theory Let µ be a nite Borel measure on R and suppose that K := supp(µ) is compact. How can we describe lim n nλ n(µ, x) when K can be any compact set of R, not just [ 1, 1]? To do this, we need concepts from potential theory. Denition (energy) Let µ be an arbitrary Borel measure on the complex plane. The energy of µ is dened as 1 I (µ) := log z w dµ(z)dµ(w). The energy of the set K is dened as I (K) := inf I (µ), µ M 1(K) where M 1 (K) denotes the probability measures supported on K. T. Danka (University of Szeged) Christoel functions / 37

16 Equilibrium measure Let K C be a set with I (K) >. Denition (equilibrium measure) If ν K is a probability measure supported on K such that I (ν K ) = I (K), then ν K is called an equilibrium measure of K. If K is compact then there is a unique equilibrium measure. Example 1. The equilibrium measure for the unit circle T is the normed Lebesgue measure dν T (e it ) = 1 2π dt, t [ π, π). Example 2. The equilibrium measure for the interval [ 1, 1] is the Chebyshev-distribution 1 dν [ 1,1] (x) = π dx, x [ 1, 1]. 2 1 x T. Danka (University of Szeged) Christoel functions / 37

17 When the support is a compact set of R Theorem (Totik, 2000 and 2008) Let µ be a nite Borel measure on R regular in the sense of Stahl-Totik supported on a compact set K := supp(µ). Suppose that on some interval I K the measure is absolutely continuous with dµ(x) = w(x)dx, where w is positive and continuous on I. Then lim nλ n(µ, x) = dµ(x) n dν K holds for x I uniformly in compact subsets of I, where ν K is the equilibrium measure for K. T. Danka (University of Szeged) Christoel functions / 37

18 When the support is a compact set of R Theorem (Totik, 2000 and 2008) Let µ be a nite Borel measure on R regular in the sense of Stahl-Totik supported on a compact set K := supp(µ). Suppose that on some interval I K the measure is absolutely continuous with dµ(x) = w(x)dx, where w is positive and continuous on I. Then lim nλ n(µ, x) = dµ(x) n dν K holds for x I uniformly in compact subsets of I, where ν K is the equilibrium measure for K. Remark. The reason for "2000 and 2008" is that in the 2000 version of the theorem an additional assumption about K was required, but it was dropped in T. Danka (University of Szeged) Christoel functions / 37

19 The polynomial inverse image method of Totik The idea of Totik is the following. Let T N be a polynomial such that T N is of degree N, all of its zeros are real and simple, and if x 0 is a local extrema of T N, then T N (x) 1. Let E := T 1 N ([ 1, 1]). The main idea can be summarized in the following steps. Prove asymptotics for Christoel functions with respect to nice measures supported on E = T 1 N ([ 1, 1]). Approximate general compact sets with sets of form T 1 N ([ 1, 1]), then prove asymptotics for Christoel functions with respect to measures supported on that general compact set by comparing measures. Remark. The roots of the idea of using polynomial inverse images can be found in a paper of J. S. Geronimo and W. van Assche. T. Danka (University of Szeged) Christoel functions / 37

20 The image is from the paper "Orthogonal Polynomials on Several Intervals Via a Polynomial Mapping" by J. S. Geronimo and W. van Assche. T. Danka (University of Szeged) Christoel functions / 37

21 The comparison method Comparing measures, one can establish limits using known results for "nice" measures. Theorem (Totik, 2000) Let µ 1 and µ 2 be two nite Borel measures regular in the sense of Stahl-Totik supported on [ 1, 1]. Suppose that there is an open interval I [ 1, 1] such that the measures coincide there. Then and lim sup n nλ n (µ 1, x) = lim sup nλ n (µ 2, x), n lim inf n nλ n(µ 1, x) = lim inf n nλ n(µ 2, x), x I x I T. Danka (University of Szeged) Christoel functions / 37

22 The comparison method Comparing measures, one can establish limits using known results for "nice" measures. Theorem (Totik, 2000) Let µ 1 and µ 2 be two nite Borel measures regular in the sense of Stahl-Totik supported on [ 1, 1]. Suppose that there is an open interval I [ 1, 1] such that the measures coincide there. Then and lim sup n nλ n (µ 1, x) = lim sup nλ n (µ 2, x), n lim inf n nλ n(µ 1, x) = lim inf n nλ n(µ 2, x), x I x I Remark. This method was improved by D. S. Lubinsky in He used Christoel functions to prove so-called universality limits which appear in the theory of random matrices. T. Danka (University of Szeged) Christoel functions / 37

23 Proof (as a demonstration of the use of regularity) Proof. µ 1, µ 2 measures on [ 1, 1], coincide on ( δ, δ). Suppose: P n is extremal for λ n (µ 1, 0), i.e. P n 2 L = λ 2(µ 1) n(µ 1, 0) and P n (0) = 1. Dene Pn (x) := P n (x) (1 x ) 2 ηn, 2 where η > 0 arbitrary. Then λ n+2 ηn (µ 2, 0) = 1 1 δ δ P n (x) 2 dµ 2 (x) ( δ Pn (x) 2 dµ 2 (x) δ ) P n (x) 2 dµ 2 (x). T. Danka (University of Szeged) Christoel functions / 37

24 On one hand, since we have (1 x ) 2 ηn 1, x [ 1, 1], 2 δ δ P n (x) 2 dµ 2 (x) = δ δ δ δ 1 1 = λ n (µ 1, 0). P n (x) 2 dµ 1 (x) P n (x) 2 dµ 1 (x) P n (x) 2 dµ 1 (x) T. Danka (University of Szeged) Christoel functions / 37

25 On the other hand, µ 1 Reg says that lim sup n ( ) 1/n Pn [ 1,1] 1. P n L2 (µ 1) (No cheating here: since [ 1, 1] is rather special, we can write P n [ 1,1] rather then P n (z), z [ 1, 1].) This implies that for arbitrary ε > 0 we have P n [ 1,1] (1 + ε) n P n L2(µ 1) (1 + ε) n M, if n is large. (Since P n is extremal for λ n (µ 1, 0), P n L2(µ 1) M holds for some M > 0.) T. Danka (University of Szeged) Christoel functions / 37

26 Using these, we obtain ( δ If ε > 0 is so small such that δ ) P n (x) 2 dµ 2 (x) ) ηn Pn (1 δ2 [ 1,1] M 2 ) ηn (1 (1 δ2 + ε) n M. 2 ) ηn 1 (1 δ2 P n (x) 2 dµ 1 (x) 2 1 then for some q < 1. ( δ + 1 ) η(1 (1 δ2 + ε) < 1, 1 δ 2 ) P n (x) 2 dµ 2 (x) = O(q n ) T. Danka (University of Szeged) Christoel functions / 37

27 This way, we obtain that λ n+2 ηn (µ 2, 0) λ n (µ 1, 0) + O(q n ). Since lim sup n (n + 2 ηn )λ n+2 ηn (µ 2, 0) = lim sup n nλ n (µ 2, 0) (easy to prove), we have lim sup n nλ n (µ 2, 0) (1 + 2η) lim sup nλ n (µ 1, 0). n Since η > 0 was arbitrary and the roles of µ 1 and µ 2 can be interchanged, we have lim sup n nλ n (µ 2, 0) = lim sup nλ n (µ 1, 0). n The equality with lim inf can be done similarly. T. Danka (University of Szeged) Christoel functions / 37

28 Jacobi measures What happens when the weight function of µ is not continuous or not strictly positive in a neighbourhood of the point? For example, dµ(x) = (1 x)α (1 + x)β dx, x [ 1, 1] or dµ(x) = x α dx, x [ 1, 1] T. Danka (University of Szeged) Christo el functions / 37

29 Jacobi measures Theorem (Nevai, 1979) Let µ be a Borel measure supported on [ 1, 1] given by where α, β > 1. Then Three questions. dµ(x) = (1 x) α (1 + x) β dx, x [ 1, 1], lim n n2α+2 λ n (µ, 1) = 2 α+1 Γ(α + 1)Γ(α + 2). What happens when the measure is modied? What happens on more general sets? What if the singularity is inside? T. Danka (University of Szeged) Christoel functions / 37

30 Modied Jacobi measures A. B. J. Kuijlaars, M. Vanlessen (2001): Let dµ(x) = (1 x) α (1 + x) β h(x)dx, x [ 1, 1], α, β > 1, h is real analytic and strictly positive. Then lim n n2α+2 λ n (µ, 1) = h(1)2 α+1 Γ(α + 1)Γ(α + 2). D. S. Lubinsky (2008): Let µ be a measure regular in the sense of Stahl-Totik supported on [ 1, 1] and suppose that it is absolutely continuous in [1 δ, 1] with dµ(x) = (1 x) α w(x)dx, x [1 δ, 1], α > 1, w is continuous and strictly positive at 1. Then lim n n2α+2 λ n (µ, 1) = w(1)2 α+1 Γ(α + 1)Γ(α + 2). Remark. They both proved much stronger results. Kuijlaars and Vanlessen used Riemann-Hilbert methods while Lubinsky used comparison methods. T. Danka (University of Szeged) Christoel functions / 37

31 Jacobi measures on Jordan curves and arcs I Let µ be a nite Borel measure on the complex plane and assume that µ is regular in the sense of Stahl and Totik, the support γ := supp(µ) consists of nitely many Jordan curves and arcs lying exterior to each other. Suppose that z 0 is in the two dimensional interior of γ and γ is C 2 -smooth in a neighbourhood of z 0, dµ(z) = w(z) z z 0 α ds γ (z), α > 1 for some continuous and positive weight w(z) on that neighbourhood, where s γ denotes the arc-length measure of γ. Theorem (Totik, D., 2015) With the previous notations and conditions, we have w(z 0 ) ( α + 1 lim n nα+1 λ n (µ, z 0 ) = (πω γ (z 0 )) α+1 2α+1 Γ 2 where ω γ denotes the density of the equilibrium measure for γ. )( α ), T. Danka (University of Szeged) Christoel functions / 37

32 Jacobi measures on Jordan curves and arcs I z 0 A typical situation where the previous theorem can be applied T. Danka (University of Szeged) Christoel functions / 37

33 Jacobi measures on Jordan curves and arcs II Now suppose that the support of µ consists of nitely many Jordan curves and arcs lying exterior to each other and z 0 is an endpoint for one of the arcs. Theorem (Totik, D., 2015) With the previous notations and conditions, we have where w(z 0 ) lim n n2α+2 λ n (µ, z 0 ) = Γ(α + 1)Γ(α + 2), 2α+2 (πm(γ, z 0 )) M(γ, z 0 ) := lim z z0 ω γ (z 0 ) z z 0,z γ is the behaviour of the equilibrium density near the endpoint z 0. T. Danka (University of Szeged) Christoel functions / 37

34 Jacobi measures on Jordan curves and arcs II z 0 A typical situation where the previous theorem can be applied T. Danka (University of Szeged) Christoel functions / 37

35 Idea of proof when there are no arcs Prove it for dµ(e it ) = e it i α dt, t [ π, π). Prove it for measures supported on level lines of polynomials. Approximate Jordan curves with level lines of polynomials, then use comparison method to prove the theorem. T. Danka (University of Szeged) Christoel functions / 37

36 Idea of proof when there are no arcs How to approximate Jordan curves with level lines of polynomials? Theorem (Nagy, Totik, 2005) Let Γ consist nitely many Jordan curves lying exterior to each other, let P γ be arbitrary, and suppose that Γ is C 2 smooth in a neighbourhood of P. Then for every ε > 0 there is a polynomial T N (z) such that the level line σ := {z : T N (z) = 1} touches Γ at P and every component of Γ contains in its interior precisely one component of σ and ω σ (P) ω Γ (P) + ε, where ω Γ and ω σ denotes the equilibrium density of Γ and σ respectively. (Similarly, Γ can be approximated from the inside.) T. Danka (University of Szeged) Christoel functions / 37

37 Idea of proof when there are no arcs Γ and the approximating curve σ. Image taken from preprint T. Danka, V. Totik, Christoel functions with power type weights. T. Danka (University of Szeged) Christoel functions / 37

38 Idea of proof when there is only one arc Let J be a Jordan arc, 0 J. Main idea: Divide up J into n subarcs I k = a k b k, k {1, 2,..., n} around 0 using the zeros of a Bessel function. Divide up J into n subarcs Ik = ak b k, k {1, 2,..., n} using a discretization of the equilibrium measure and take the center of mass ξ k of I k. Using a k and ξ k as zeros, conjure up a polynomial P n which is almost extremal, then estimate λ n (µ, 0), where dµ(z) = z α ds J (z). T. Danka (University of Szeged) Christoel functions / 37

39 Future work It would be nice to know: does something stronger, like λ n (µ, z 0 + a/n) lim = 1 n λ n (µ, z 0 ) holds uniformly for a in compact subsets of C? (These type of limits are strongly connected to the so-called universality limits in random matrix theory.) Work in progress! It would be very nice to know: can the Szeg condition π π log w(eit )dt > be replaced in the Máté-Nevai-Totik theorem for w(e it ) > 0 Lebesgue-almost everywhere? That is, does lim nλ n(µ, e it ) = 2πw(e it ), n t [ π, π) almost everywhere hold with the much more weaker w(e it ) > 0 a.e. condition? T. Danka (University of Szeged) Christoel functions / 37

40 References I P. Nevai Orthogonal polynomials Memoirs of the AMS, no. 213 (1979) A. Máté, P. Nevai, V. Totik Szeg 's extremum problem on the unit circle. Annals of Mathematics, Vol. 134, No. 2 (1991) V. Totik Asymptotics for Christoel functions for general measures on the real line Journal D'Analyse Mathématique, Vol. 81 (2000) A. B. J. Kuijlaars and M. Vanlessen Universality for eigenvalue correlations from the modied Jacobi unitary ensemble Int. Math. Res. Not., Vol. 30 (2002) T. Danka (University of Szeged) Christoel functions / 37

41 References II D. S. Lubinsky A new approach to universality limits involving orthogonal polynomials Annals of Mathematics, Vol. 170 (2009) D. S. Lubinsky Universality limits at the hard edge of the spectrum for measures with compact support Int. Math. Res. Not., Art. ID rnn 099 (2008) V. Totik Universality and ne zero spacing on general sets Arkiv för Matematik, Vol. 47 (2009) V. Totik Christoel functions on curves and domains Transactions of the AMS, Vol. 362 (2010) T. Danka (University of Szeged) Christoel functions / 37

42 References III V. Totik Asymptotics of Christoel functions on arcs and curves Advances in Mathematics, Vol. 252 (2014) T. Danka and V. Totik Christoel functions with power type weights submitted for consideration for publication T. Danka (University of Szeged) Christoel functions / 37

43 Thank you for your attention! T. Danka (University of Szeged) Christoel functions / 37